Laue symmetry

No systematic extinctions were found in addition to those characteristic of body-centering. The only space groups with Laue symmetry Th allowed by this observation are T, T3, and T6. No non-systematic absences were recorded. 

The Laue symmetry of the diffraction pattern is now reduced to the symmetry of the non-centrosymmetric point group to which the crystal belongs (see Table 2 a listing of symmetry-equivalent reflections in the non-centrosymmetric point groups is available26). Under these circumstances, it is possible to determine different manifestations of non-centrosymmetry in a crystal, such as ... 

The Friedel pairs are a subset of the so-called Bijvoet pairs, i.e., pairs of any two reflections which are equivalent by Laue symmetry but not by the point group symmetry of the crystal (see Table 2 a full listing of these reflections is available4 1). All Bijvoet pairs may be used for a comparison, but since the measured intensity differences between general Bijvoet pairs are more likely to be affected by systematic error, the conclusion based on their comparison may be less reliable than in the case of Friedel pairs. 

Since the diffraction symmetry is shown so strikingly in Laue photographs, it is often called the Laue symmetry . The information on diffraction symmetry is of course all contained in moving-crystal photographs taken by monochromatic X-rays, provided that reflections with similar indices are separated, as they are in tilted crystal photographs... 

The information obtainable from the Laue symmetry is meagre it consists simply in the distinction, between crystal classes, and then only in the more symmetrical systems—cubic, tetragonal, hexagonal, and trigonal (see Table VI). But it is useful in cases in which morphological features do not give clear evidence on this point. 

System Symmetry Symmetry Point Groups (Laue) Symmetry... 

The term diffraction symmetry, which is often called Laue symmetry because it can be most conspicuous in a type of X-ray photograph called a Laue photograph, is applied to those point groups that are recognizably different in the diffraction pattern. 

Distinguishing Space Groups by Systematic Absences. From the symmetry and metric properties of an X-ray diffraction pattern we can determine which of the 6 crystal systems and, further, which of the 11 Laue symmetries we are dealing with. Since we need to know the specific space group in order to solve and refine a crystal structure, we would still be in a highly unsatisfactory situation were it not for the fact that the X-ray data can tell us still more. 

Laue symmetry is 2/m (disregarding the translational components of the 21-axis and the c glide), so the space group belongs to the monoclinic system. The simplified symbol P2 /c normally replaces the full symbol P 2 /c, and the old Schonflies symbol C A (the fifth space group derived from the point group C2h) is seldom used. 

Table III. 1. Crystal class, Laue symmetry, and number of possible different orientations of the electric field gradient tensor for the nuclei at the general point position in the crystal lattice...

Crystal class Laue symmetry Number of different orientations of the EFG tensor for the general point position... 

Crystal system Bravais lattices Unit cell constraints Laue symmetry... 

Crystal system (7) Characteristic symmetry Lattice [Laue) symmetry Axial and angular constraints from symmetry ... 

Any symmetry in the intensities in the diffraction pattern other than that implied by Friedel s Law is called Laue symmetry (because it can be displayed on Laue X-ray diffraction photographs of an appropriately-aligned crystal, see Figure 4.16). Friedel s Law implies that there is a center of symmetry in the diffraction pattern. Therefore the Laue symmetry-displayed by the diffraction pattern is the point-group symmetry of the crystal with an additional center of symmetry (if this does not already exist). If a crystal is monoclinic then, the intensities I[hkl) and I hkl) are the same, although I hkl) does not equal I[hkl). Orthorhombic... 

Laue symmetry of an orthorhombic crystal (see Figure 4.17). Thus, Laue symmetry, not unit cell dimensions, give a measure of the crystal system. As was stressed earlier, it is the symmetry of the diffraction pattern that tells us that a crystal (lattice) is orthorhombic, not the fact that a = /9 = 7 = 90°. 

FIGURE 4.16. Laue symmetry. A diffraction photograph of tetragonal lysozyme, viewed down its unique axis (c). Note the fourfold symmetry of the diffraction pattern. 

Laue symmetry Symmetry in the intensities of the diffraction pattern beyond that expected from Friedel s Law. The Laue symmetry of the diffraction pattern of a crystal is the point-group symmetry of the crystal plus, as Friedel noted, a center of symmetry. There are 11 Laue symmetry groups. 

The symmetry of the Patterson function is the same as the Laue symmetry of the crystal. The Patterson function for space groups that have symmetry operations with translational components (screw axes and glide planes) has an added property that is very useful for the determination of the coordinates of heavy atoms. Specific peaks, first described b David Harker, are associated with the vectors between atoms related by these symmetry operators. These peaks are found along lines or sections (Figure 8.17). For example, in the space group P2i2i2i there are atoms at... 

What is the symmetry of the reciprocal lattice That is, what are the symmetry operators that relate sets of identical intensities The symmetry group that we observe for a crystal in reciprocal space, namely the diffraction pattern symmetry, is called the Laue symmetry, or Laue group. 

For macromolecular crystals, the symmetry of the diffraction pattern (the Laue symmetry) must be generated by Friedel s law, plus the rotational components of symmetry axes present in the crystal. Once the rotational elements have been identified, it is necessary to deduce whether they are pure rotational operators or some sort of screw axes. For dyads,... 

To single out the correct space group among those showing the same Laue symmetry the systematically absent reflections have to be studied. Since the structure factor ... 

Space Group Determination Initially, the space group was determined from normally exposed photographic films. Extinctions noted were hkly none hkO, h odd Okl, k + l odd hOJ, none. These are indicative of space groups Pnma or Pn2t a, since the Laue symmetry is mmm. There are four formula units in the cell so that no symmetry need be imposed on the cation in Pn2t a, whereas in Pnma either a mirror plane or a center of symmetry is imposed. 

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